Alternating complexity of counting first-order logic for the subword order

Abstract

This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by threshold counting quantifiers. The main result shows that the two-variable fragment of this logic can be decided in twofold exponential alternating time with linearly many alternations (and therefore in particular in twofold exponential space as announced in the conference version (Kuske and Schwarz, in: MFCS’20, Leibniz International Proceedings in Informatics (LIPIcs) vol. 170, pp 56:1–56:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020) of this paper) provided the regular predicates are restricted to piecewise testable ones. This result improves prior insights by Karandikar and Schnoebelen by extending the logic and saving one exponent in the space bound. Its proof consists of two main parts: First, we provide a quantifier elimination procedure that results in a formula with constants of bounded length (this generalises the procedure by Karandikar and Schnoebelen for first-order logic). From this, it follows that quantification in formulas can be restricted to words of bounded length, i.e., the second part of the proof is an adaptation of the method by Ferrante and Rackoff to counting logic and deviates significantly from the path of reasoning by Karandikar and Schnoebelen.